论文标题
由四阶变形和相位场模型控制的多斑外壳的等几何连续性约束
Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models
论文作者
论文摘要
这项工作提出了数值技术，以在三个不同的问题类别上对多斑表面上的连续性约束强制执行。第一个涉及对薄壳的结构分析，这些壳被一般的Kirchhoff-Love运动学描述。他们的管理方程是一个矢量值，四阶，非线性，部分微分方程（PDE），需要在基于位移的有限元配方中至少$ c^1 $ continuity。第二类是由相场建模的表面相分离。他们的管理方程是Cahn -Hilliard方程 - 标量，第四阶，非线性PDE-可以耦合到薄壳PDE。第三类是由相位场方法建模的脆性断裂过程。在这项工作中，这些用标量，四阶，非线性PDE描述，类似于Cahn-Hilliard方程，并且也与薄壳PDE耦合。使用直接有限元离散化，两个相字段方程也至少需要一个$ C^1 $ - 连续公式。因此，通常由多个贴片组成的等几何表面离散化需要强制执行位移和相位场的$ C^1 $ - 接触性的约束。为此，提出了两种数值策略：为此，提出了两种数值策略：Lagrange乘数公式和惩罚方法。包括几何约束的曲线壳模型取自Duong等人。 （2017年），并扩展到Zimmermann等人的薄外壳上的耦合相位场问题进行建模。 （2019）和Paul等。 （2020）在多斑点上。考虑变形的壳，不断发展的表面上的相位分离以及薄壳的动态脆性断裂来说明它们的准确性和收敛性。
This work presents numerical techniques to enforce continuity constraints on multi-patch surfaces for three distinct problem classes. The first involves structural analysis of thin shells that are described by general Kirchhoff-Love kinematics. Their governing equation is a vector-valued, fourth-order, nonlinear, partial differential equation (PDE) that requires at least $C^1$-continuity within a displacement-based finite element formulation. The second class are surface phase separations modeled by a phase field. Their governing equation is the Cahn-Hilliard equation - a scalar, fourth-order, nonlinear PDE - that can be coupled to the thin shell PDE. The third class are brittle fracture processes modeled by a phase field approach. In this work, these are described by a scalar, fourth-order, nonlinear PDE that is similar to the Cahn-Hilliard equation and is also coupled to the thin shell PDE. Using a direct finite element discretization, the two phase field equations also require at least a $C^1$-continuous formulation. Isogeometric surface discretizations - often composed of multiple patches - thus require constraints that enforce the $C^1$-continuity of displacement and phase field. For this, two numerical strategies are presented: For this, two numerical strategies are presented: A Lagrange multiplier formulation and a penalty method. The curvilinear shell model including the geometrical constraints is taken from Duong et al. (2017) and it is extended to model the coupled phase field problems on thin shells of Zimmermann et al. (2019) and Paul et al. (2020) on multi-patches. Their accuracy and convergence are illustrated by several numerical examples considering deforming shells, phase separations on evolving surfaces, and dynamic brittle fracture of thin shells.
